On sets W \subseteq \mathbb{N} such that the infinity of W is equivalent to the existence in W of an element that is greater than a threshold number computed with using the definition of W
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2018
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Abstract
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n \geq 2. For a positive integer n, let \Theta_n denote the statement: if a system S \subseteq {x_i!=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n greater than 1, then each such solution (x_1,...,x_n) satisfies min(x_1,...,x_n) \leq f(n). The statement \Theta_9 proves that if there exists an integer x>f(9) such that x^2+1 (alternatively, x!+1) is prime, then there are infinitely many primes of the form n^2+1 (respectively, n!+1). The statement \Theta_{16} proves that if there exists a twin prime greater than f(16)+3, then there are infinitely many twin primes. We formulate the statements \Phi_n and prove: \Phi_4 equivalently expresses that there are infinitely many primes of the form n!+1, \Phi_6 implies that for infinitely many primes p the number p!+1 is prime, \Phi_6 implies that there are infinitely many primes of the form n!-1, \Phi_7 implies that there are infinitely many twin primes.